Question

find the number of triangles that can be formed using 14 points in aplane such that 4 points are colliner​

Answer 1

$\large\underline{\sf{Solution-}}$

We know,

★ If there are n non collinear points in the plane, then number of triangles formed is given by

$\rm :\longmapsto\:Number \: of \: triangles = \: ^nC_3$

and we know,

$\rm :\longmapsto\:^nC_r \: = \: \dfrac{n!}{r!(n-r)!}$

Now,

★ It is given that, 14 points in a plane out of which 4 points are collinear.

★ Number of triangles formed by joining 14 points taken 3 at a time is

$\rm :\longmapsto\:Number \: of \: triangles = \: ^{14}C_3$

$\rm \: \: = \: \dfrac{14!}{3!(14 - 3)!}$

$\rm \: \: = \: \dfrac{14!}{3! \: \: 11!}$

$\rm \: \: = \: \dfrac{14 \times 13 \times 12 \times 11!}{3 \times 2 \times 1\: \: 11!}$

$\rm \: \: = \: 28 \times 13$

$\rm \: \: = \: 364$

Now,

★ Number of triangles formed by joining 4 points taken 3 at a time is

$\rm :\longmapsto\:Number \: of \: triangles = \: ^{4}C_3$

$\rm \: \: = \: \dfrac{4!}{3!(4 - 3)!}$

$\rm \: \: = \: \dfrac{4 \times 3!}{3! \: \: 1!}$

$\rm \: \: = \: 4$

★ But 4 points are collinear which don't form a triangle when taken 3 at a time.

So,

★ Required number of triangles = 364 - 4 = 360.

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Additional Information :-

$\rm :\longmapsto\:^nC_r \: = \: \dfrac{n}{r} \: ^{n - 1}C_{r - 1}$

$\rm :\longmapsto\:^nC_0 = ^nC_n = 1$

$\rm :\longmapsto\:^nC_1 = ^nC_{n - 1} = n$

$\rm :\longmapsto\:^nC_r \: + \: ^nC_{r - 1} \: = \: ^{n + 1}C_r$

$\rm :\longmapsto\:\dfrac{^nC_r}{^nC_{r - 1}} = \dfrac{n - r + 1}{r}$